Gradient stability of high-order BDF methods and some applications

Anass Bouchriti , Morgan Pierre (1) , Nour Eddine Alaa

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 7348]

It is well-known that the backward differentation formulae (BDF) of order 1, 2 and 3 are gradient stable. This means that when such a method is used for the time discretization of a gradient flow, the associated discrete dynamical system exhibit properties similar to the continuous case, such as the existence of a Lyapunov functional. By means of a Lojasiewicz-Simon inequality, we prove convergence to equilibrium for the 3-step BDF scheme applied to the Allen-Cahn equation with an analytic nonlinearity. By introducing a notion of quadratic-stability, we also show that the BDF methods of order 4 and 5 are gradient stable, and that the k-step BDF schemes are not gradient stable for k ≥ 7. Some numerical simulations illustrate the theoretical results.

Mots clés: BDF methods; gradient flow; Allen-Cahn equation; Lojasiewicz-Simon inequality

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Optimisation et contrôle [math.OC]

Anass Bouchriti, Morgan Pierre, Nour Eddine Alaa. Gradient stability of high-order BDF methods and some applications. 2018. ⟨hal-01818370⟩